As a testament to Moonlighting's quality it was nominated for many awards, and even won a few. Here's a list of what I could find searching the imdb and emmy websites.
Won ASCAP AWARD TOP TV SERIES
Alf Clausen
Lee Holdridge
Nominated ASC AWARD OUTSTANDING ACHEIVEMENT IN CINEMATOGRAPHY IN REGULAR SERIES'
Gerald Perry Finnerman
Nominated ASC AWARD OUTSTANDING ACHEIVEMENT IN CINEMATOGRAPHY IN REGULAR SERIES'
Gerald Perry Finnerman
Nominated ARTIOS BEST CASTING FOR TV, Dramatic Episodic
Reuben Cannon (pilot)
Carol L. Dudley (series)
Won DGA AWARD OUTSTANDING DIRECTORIAL ACHEIVEMENT IN DRAMATIC SHOWS - Night- For episode "Atomic Shakespeare".
Skip Beaudine (first assistant director) (plaque)
Richard Learman (unit production manager) (plaque)
Will Mackenzie (I)
Leigh A. Webb (second assistant director) (plaque)
Won DGA AWARD OUTSTANDING DIRECTORIAL ACHEIVEMENT IN DRAMATIC SHOWS - Night- For episode "My Fair David".
Skip Beaudine (first assistant director) (plaque)
Will Mackenzie (I)
F.A. Miller (unit production manager) (plaque)
Leigh Webb (second assistant director) (plaque)
NOMINATED EDGAR BEST TELEVISION EPISODE- "The Dream Sequence Alwas Rings Twice".
Debra Frank
Carl Sautter
WON OUTSTANDING ART DIRECTION FOR A SERIES-A Womb With A View
James J. Agazzi Production Designer
Bill Harp Set Decorator
NOMINATED OUTSTANDING ACHIEVEMENT IN MUSIC COMPOSITION FOR A SERIES (dramatic underscore) -A Womb With A View
Alf Clausen Composer
NOMINATED OUTSTANDING ACHIEVEMENT IN MUSIC DIRECTION-A Womb With A View
Alf Clausen Music Director/Principal Arranger
Hummie Mann Principal Arrange
Brad Dechter Principal Arranger
George Gaffney Dance Arranger
D'Vaughn Pershing Dance Arranger
NOMINATED OUTSTANDING ACHIEVEMENT IN MUSIC COMPOSITION FOR A SERIES (dramatic underscore) -Here's Living with you, Kid
Alf Clausen Composer
NOMINATED OUTSTANDING ACHIEVEMENT IN MUSIC DIRECTION -Cool Hand Dave, Part II
Alf Clausen Music Director
Alf Clausen Principal Arranger
Hummie Mann Principal Arranger
Brad Dechter Principal Arranger
Don Nemitz Principal Arranger
George Gaffney Principal Arranger
NOMINATED OUTSTANDING ART DIRECTION FOR A SERIES- Here's Living with you, Kid
James J. Agazzi Production Designer
Bill Harp Set Decorator
NOMINATED OUTSTANDING CINEMATOGRAPHY FOR A SERIES-Here's Living with You, Kid
Gerald Perry Finnerman ASC
NOMINATED OUTSTANDING COSTUME DESIGN FOR A SERIES -Here's Living with you, Kid
Robert Turturice Costume Designer
NOMINATED OUTSTANDING GUEST PERFORMER IN A DRAMA SERIES -Los Dos Dipestos
Imogene Coca
WON OUTSTANDING ACHIEVEMENT IN HAIRSTYLING FOR A SERIES -Atomic Shakespeare
Kathryn Blondell Ms. Shepherd's Hairstylist
Josee Normand Hairstylist
WON OUTSTANDING COSTUME DESIGN FOR A SERIES -Atomic Shakespeare
Robert Turturice Costume Designer
WON OUTSTANDING EDITING FOR A SERIES (single camera production) -Atomic Shakespeare
Roger Bondelli Editor,
Neil Mandelberg Editor
WON OUTSTANDING LEAD ACTOR IN A DRAMA SERIES
Bruce Willis
NOMINATED OUTSTANDING ACHIEVEMENT IN CHOREOGRAPHY -Big Man on Mulberry Street
Bill Landrum Choreographer
Jacqui Landrum Choreographer
NOMINATED OUTSTANDING ACHIEVEMENT IN MUSIC COMPOSITION FOR A SERIES (dramatic underscore) -Atomic Shakespeare
Alf Clausen Composer
NOMINATED OUTSTANDING ART DIRECTION FOR A SERIES -Atomic Shakespeare
James J. Agazzi Production Designer
Bill Harp Set Decorator
2 NOMINATIONS OUTSTANDING DIRECTING IN A DRAMA SERIES -
I Am Curious... Maddie -Alan Arkush Director
Atomic Shakespeare-Will MacKenzie Director
NOMINATED OUTSTANDING DRAMA SERIES -
Jay Daniel Co-Executive Producer
Artie Mandelberg Supervising Producer
Karen Hall Supervising Producer
Ron Osborn Producer
Jeff Reno Producer
Roger Director Producer
Glenn Gordon Caron Executive Producer
NOMINATED OUTSTANDING SOUND MIXING FOR A DRAMA SERIES - Atomic Shakespeare
Martin Raymond Bolger Sound Mixer
Dave Hudson Sound Mixer
Mel Metcalf Sound Mixer
Terry Porter Sound Mixer
NOMINATED OUTSTANDING SUPPORTING ACTRESS IN A DRAMA SERIES -
Allyce Beasley
3 NOMINATIONS OUTSTANDING WRITING IN A DRAMA SERIES -
I am Curious...Maddie-Jeff Reno Teleplay, Ron Osborn Story, Karen Hall Story, Roger Director Story, Charles H. Eglee Story
Atomic Shakespeare ï Ron Osborn Writer, Jeff Reno Writer
I am Curious...Maddie -Glenn Gordon Caron Teleplay
WON OUTSTANDING EDITING FOR A SERIES (single camera production) -
The Dream Sequence Always Rings Twice-Neil Mandelberg Editor
ALSO NOMINATED Every Daughter's Father is a Virgin- Roger Bondelli Editor
NOMINATED OUTSTANDING ACHIEVEMENT IN HAIRSTYLING FOR A SERIES - The Dream Sequence Always Rings Twice
Judy Crown Hairstylist
Josee Normand Hairstylist
NOMINATED OUTSTANDING ACHIEVEMENT IN MUSIC COMPOSITION FOR A SERIES (dramatic underscore) - The Dream Sequence Always Rings Twice
Alf Clausen Composer
NOMINATED OUTSTANDING ART DIRECTION FOR A SERIES - The Dream Sequence Always Rings Twice
James J. Agazzi Production Designer
Bill Harp Set Decorator
NOMINATED OUTSTANDING CINEMATOGRAPHY FOR A SERIES -The Dream Sequence Always Rings Twice
Gerald Perry Finnerman A.S.C.
NOMINATED OUTSTANDING COSTUME DESIGN FOR A SERIES - The Dream Sequence Always Rings Twice
Robert Turturice Costume Designer
2 NOMINATIONS OUTSTANDING DIRECTING IN A DRAMA SERIES
My Fair David- Will MacKenzie Director
The Dream Sequence Always Rings Twice-Peter Werner Director
NOMINATED OUTSTANDING DRAMA SERIES
Glenn Gordon Caron Executive Producer
Jay Daniel Co-Executive Producer
Artie Mandelberg Supervising Producer
Jay Daniel Producer
Ron Osborn Producer
Jeff Reno Producer
NOMINATED OUTSTANDING GUEST PERFORMER IN A DRAMA SERIES -Camille
Whoopi Goldberg
NOMINATED OUTSTANDING LEAD ACTOR IN A DRAMA SERIES
Bruce Willis
NOMINATED OUTSTANDING LEAD ACTRESS IN A DRAMA SERIES
Cybill Shepherd
NOMINATED OUTSTANDING SUPPORTING ACTRESS IN A DRAMA SERIES
Allyce Beasley
2 NOMINATIONS OUTSTANDING WRITING IN A DRAMA SERIES
Twas The Episode Before Christmas-Glenn Gordon Caron Writer
The Dream Sequence Always Rings Twice - Debra Frank Writer, Carl Sautter Writer
NOMINATED OUTSTANDING ACHIEVEMENT IN MUSIC AND LYRICS -Pilot
Lee Holdridge Composer
Al Jarreau Lyricist
NOMINATED OUTSTANDING DIRECTING IN A COMEDY SERIES -Pilot
Robert Butler Director
Nominated Golden Globe Best Performance by an Actor in a TV-Series - Comedy/Musical
Bruce Willis
Nominated Best Performance by an Actress in a Supporting Role in a Series, Mini-Series or Motion Picture Made for TV
Allyce Beasley
Nominated Best Performance by an Actress in a TV-Series - Comedy/Musical
Cybill Shepherd
Nominated Best TV-Series - Comedy/Musical
Won Golden Globe Best Performance by an Actor in a TV-Series - Comedy/Musical
Bruce Willis
Won Best Performance by an Actress in a TV-Series - Comedy/Musical
Cybill Shepherd
Nominated Golden Globe Best TV-Series - Comedy/Musical
Won Golden Globe Best Performance by an Actress in a TV-Series - Comedy/Musical
Cybill Shepherd - Tied with Estelle Getty for "Golden Girls, The" (1985).
Nominated Golden Globe Best Performance by an Actor in a TV-Series - Comedy/Musical
Bruce Willis
Nominated Best TV-Series - Comedy/Musical